Moreover, one can look at this construction from the perspective of (finite distance) holographic dualities à la AdS/CFT correspondence. As the 2D Ising model becomes a conformal field theory (CFT) in its critical regime, the exact equivalence of the present formulation between the 3D quantum gravity and the 2D Ising partition function, which holds for every value of Ising couplings, can be understood as a non-critical version of the gauge-gravity holography. Interpreting the Ising partition function for non-critical couplings as a non-critical version of conformal blocks, the 3D Ponzano-Regge path integral is realized as gluing such 2D non-critical blocks. This version of holography holds for discrete quantized geometries and not only at the level of field theories in the continuum limit (see \(\textit {e.g.}\)  [1]}, [2]} for holographic duality at the field theory level).
More recent work following this line of thought and investigating the holographic behaviour of the Ponzano-Regge path integral can be found in [3]}, [4]}, [5]}.
A hope is that this reformulation will lead to new developments in the investigation of the phase diagram of 3D quantum gravity and the implementation of quasi-local holography in spinfoams and loop-gravity-inspired path integrals for quantum gravity.
